Arithmetic means of crop and weed biomass (g/m\(^{-2}\)) are in Figure 1.

Figure 1: Arithmetic means of crop and weed biomass.

Results

Crop biomass was the strongest factor affecting weed biomass (Table 2 so a non-linear regression was fitted for crop - weed competition (Figure 2). Collard’s weed suppression was distinctive among all the treatments (Table 4.

A nonlinear regression was fitted with \(Y = \frac{C}{1 + I_w B_c}\), where

\(Y\) is predicted weed biomass,
\(C\) is weed biomass in the control treatment,
\(I_w\) is an estimated coefficient,
\(B_c\) is crop biomass in cover crop treatment

## 
## Formula: weed.biomass.g.per.sq.m ~ 148/(1 + i.w * crop.biomass.g.per.sq.m)
## 
## Parameters:
##     Estimate Std. Error t value Pr(>|t|)  
## i.w 0.008078   0.003155   2.561   0.0137 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 75.39 on 47 degrees of freedom
## 
## Number of iterations to convergence: 6 
## Achieved convergence tolerance: 3.688e-06

Figure 2: Weed suppression from Brassicaceae biomass

Linear model diagnosis

Weed suppression from different Brassicacaea species was comparable among each other and with the control treatments (Table 1), even though crop biomass differed by species (Table 2).

Using ggResidpanel version 0.3.0 (Goode and Rey, 2019) for model diagnosis, no predictable pattern in the plots of residuals vs. predicted values suggests that the analysis models fit the data well (Figures ?? and 5), but AIC values of the ANCOVA models were lower than those of the ANOVA models: 448 vs. 459 for crop responses in ANCOVA vs. ANOVA; and 159 vs. 174 for weed biomass responses in ANCOVA vs ANOVA.

Crop biomass in response to treatment and weed biomass

fsb.crops <- fsb.wide  %>% filter(treatment != "N") # Control treatments had no crop
crops.lm1 <- lm(crop.biomass.g.per.sq.m  ~ as.factor(block) + species.y + weed.biomass.g.per.sq.m, data = fsb.crops)


crops.lm2 <- lm(crop.biomass.g.per.sq.m  ~ as.factor(block) + species.y , data = fsb.crops)
resid_panel(crops.lm2)

Figure 3: Diagnosis plot for the ANCOVA and ANOVA models of crop biomass responses

list(crops.lm1, crops.lm2) %>% 
  lapply(resid_panel, nrow = 2) %>% 
  wrap_plots() +
  plot_annotation(tag_levels = "A")

Figure 4: Diagnosis plot for the ANCOVA and ANOVA models of crop biomass responses

# AIC(crops.lm1, crops.lm2) 448 vs 459

Weed biomass in response to treatment and crop biomass - ANCOVA

weeds.lm1 <- lm(log(weed.biomass.g.per.sq.m + 1) ~ as.factor(block) + species.y + crop.biomass.g.per.sq.m, data = fsb.wide )


weeds.lm2 <- lm(log(weed.biomass.g.per.sq.m + 1) ~ as.factor(block) + species.y , data = fsb.wide )

list(weeds.lm1, weeds.lm2) %>% 
  lapply(resid_panel, nrow = 2) %>% 
  wrap_plots() +
  plot_annotation(tag_levels = "A")

Figure 5: Diagnosis plot for the linear model of weed biomass responding to treatment and crop biomass

# AIC(weeds.lm1, weeds.lm2) 159 vs 174

Linear models results

Estimated marginal means of weed and crop biomass in each treatment are shown in Figures 6 and 7.

Figure 6: Estimated marginal means of weed biomass (back transformed from ln(x +1)). The error bars represent 1 sd on each side.

Figure 7: Estimated marginal means of crop biomass. The error bars represent 1 sd on each side.

The arrow plots below (Figures 8 and 9) convey the same information with the bar charts, but are quicker to make.

Figure 8: Estimated marginal means of weed biomass (back transformed from ln(x +1)). The black dots are the estimated marginal means. The blue bars indicate the estimated 95% confidence intervals. The arrows indicate whether two estimated means were significantly different. Overlapping arrows means non-significant difference.

Figure 9: Estimated marginal means of crop biomass. The blue bars indicate the estimated 95% confidence intervals. The black dots are the estimated marginal means. The arrows indicate whether two estimated means were significantly different. Overlapping arrows means non-significant difference.